470 DR. A. N. WHITEHEAD ON 



being subdivided into points of space and particles of matter. The essential relation 

 has for its field the points of space only. Slight variants (not considered here) can 

 be given to the concept by varying the properties of the essential relation, so as to 

 make the geometry non-Euclidean, or, retaining Euclidean geometry, so as to give 

 various forms to the essential relation and the resulting axioms. In the exposition of 

 a system of geometrical axioms for Concept I., VEBLEN'S memoir (cf. loc. cit.), to 

 which I am largely indebted, will be followed. The changes which are made from 

 VEBLEN'S treatment are (i) in the addition of the symbolism which emphasizes the 

 idea of the essential relation, and (ii) in the fact that the question of the independence 

 of axioms is here ignored, through a desire not to overload this memoir with 

 difficulties (both for the author and reader) belonging to another part of the subject. 

 As the result of (ii), some of VEBLEN'S definitions and axioms have been simplified 

 (and, in a sense, spoiled). The axioms thus obtained for Concept I. will shorten our 

 investigations of other concepts by serving as a standard of comparison to determine 

 whether the axioms of the other concepts are sufficient to yield three-dimensional 

 Euclidean geometry.* 



The essential relation (called R) is triadic. R ; (6c) means the points a, b, c are in 

 the linear order (or the Tel-order) abc. The relation R, when R'(afrc) holds, is not 

 symmetrical as between the three points a, b, and c ; namely, it will be found that a 

 and b (or b and c) cannot be interchanged. 



Definitions of Concept I. 



Definition. The class R ; (a;6) is the segment between a and b ; and the class 

 R ; (6;) is the segmental prolongation of ab beyond b ; and the class R : (;a6) is the 

 segmental prolongation of ab beyond a. It follows from the subsequent axioms that 

 R ; (a6;) is identical with R ; (;6a). 



Definition. The straight line ab is the logical sum of R ; (a;i) and R ; (;afe) and 

 R ; (a6;) together with a and b themselves. Its symbol is R ; a6. The definition in 

 symbols is 



R ; ;a6 u R ; a;& u R ; afc; u i'a u i'& Df 



* On the philosophical questions connected with the mathematical analysis of geometry cf. ' A Critical 

 Exposition of the Philosophy of LEIBNIZ,' Cambridge, 1900, and 'The Principles of Mathematics,' 

 Cambridge, 1903, both by BERTRAND KUSSELL ; and also two articles by L. COUTURAT in the 'Revue de 

 Metaphysique et de Morale' (Paris) for May and September, 1904, one entitled "La philosophic des 

 Mathematiques de KANT " and the other " Les principes des Mathe"matiques VI. La geometric"; also 

 POINCARE'S ' Science and Hypothesis,' Part II., English translation, London, 1905. 



For expositions of exact systems of axioms cf. 'Vorlesungen iiber neuere Geometric,' Leipzig, 1882, 

 by PASCH ; also ' I Principii di Geometria,' Turin, 1889, by PEANO; also " I Principii della Geometria di 

 Posizione," 'Trans. Acad. of Turin,' 1898, by PIERI; also HILBERT'S 'Foundations of Geometry,' Engl. 

 Transl., Chicago, 1902 ; also Professor E. H. MOORE, " On the Projective Axioms of Geometry," ' Transact. 

 of the Amer. Math. Soc.,' 1903; also Dr. 0. VEBLEN (loc. cit.); also Professor J. ROYCE (loc. cit.). 



